When the length of the exchange cycles is not limited there is always a stable solution that can be found efficiently by Gale's Top Trading Cycle (TTC) algorithm [1]}. Moreover, if the preferences are strict then the TTC solution is unique and strongly stable [2]}, as suggested in the seminal paper on KEPs by [3]}. However, when the length of cycles is limited, which is typically the case for KEPs, a stable or strongly stable solution may not exist and the problem of deciding its existence is NP-hard [4]}, [5]} even for tripartite graphs (also known as the cyclic 3D stable matching problem [6]}).
